We establish a relation between the Bollobás–Riordan
polynomial of a ribbon graph with the relative Tutte polynomial of a
plane graph obtained from the ribbon graph using its projection to
the plane in a nontrivial way. Also we give a duality formula for
the relative Tutte polynomial of dual plane graphs and an expression
of the Kauffman bracket of a virtual link as a specialization of the
relative Tutte polynomial.

Given a graph on a surface, we will construct a special associated
plane graph which contains all of the topological information coming
from the embedding of the graph into the surface. These constructed
plane graphs usually have some extra (distinguished) edges and extra
vertices. They are called relative plane graphs.

Definition A relative plane graph is a plane graph $G$
with a distinguished subset $H\subseteq E(G)$ of edges. The edges
$H$ are called the 0-edges of $G$. Edges in $E(G)\backslash H$ will
be referred to as regular edges.

The motivation of our work comes from knot theory. The classical
Thistlethwaite theorem ([Thistlethwaite1987]) relates the Jones polynomial of an
alternating link to the Tutte polynomial of a plane graph obtained
from a checkerboard coloring of the regions of the link diagram.
This theorem has two different kinds of generalizations to virtual
links. One ([Chmutov2009]; [Chmutov and Pak2007]; [Chmutov and Voltz2008]; [Dasbach et al.2008]; [Moffatt2010]) involves graphs on surfaces
and a topological version of the Tutte polynomial due to
Bollobás and Riordan ([Bollobás and Riordan2002]). Another generalization is
based on a relative version of the Tutte polynomial found by Diao
and Hetyei ([Diao and Hetyei2010]). In this paper we establish a direct
relation between the Bollobás–Riordan and relative Tutte
polynomials that explains how these two generalizations are
connected.

In Sect. 2 we explain the construction
of a relative plane graph from a ribbon graph as well as how to
recover a ribbon graph from a relative plane graph. Our main theorem
is formulated in Sect. 3 and proved in
Sect. 4. In Sect. 5 we describe the
relation between the relative Tutte polynomials of dual plane graphs
that generalizes the classical relation $T_{G}(x,y)=T_{G^{*}}(y,x)$.
In Sect. 6 we obtain the Kauffman bracket of a virtual
link in terms of the relative Tutte polynomial, improving the
theorem of [Diao and Hetyei2010]. Section 7 places our relation
between the Bollobás–Riordan polynomial and relative Tutte
polynomial within the context of other polynomial invariants of
graphs on surfaces.

This work has been done as a part of the Summer 2010 undergraduate
research working group

“Knots and Graphs” at the Ohio State University. We are grateful
to all participants of the group for valuable discussions and to the
OSU Honors Program Research Fund for the student financial support.

We refer to [Biggs1993], [Godsil and Royle2001], [Gross and Tucker1987], [Lando and Zvonkin2004], [Loebl2010] and [Mohar and Thomassen2001] for the standard general
notions and terminology of (topological) graph theory.

Definition 2.1.

([Bollobás and Riordan2002]) By a ribbon graph we mean an abstract (not
necessarily orientable) surface with boundary decomposed into a
number of closed topological discs of two types, vertex-discs
and edge-ribbons, satisfying the following natural conditions:
the discs of the same type are pairwise disjoint; the vertex-discs
and the edge-ribbons intersect by disjoint line segments, each such
line segment lies on the boundary of precisely one vertex and
precisely one edge, and every edge contains exactly two such line
segments, which are not adjacent.

Ribbon graphs are considered up to homeomorphisms of the underlying
surfaces preserving the decomposition.

Here are three examples.

A ribbon graph may be given by an arrow
presentation.

Definition 2.2.

([Chmutov2009]) An arrow presentation consists of a
set of disjoint circles together with a collection of arrow markings
on these circles. These arrows are labeled in pairs. To obtain a
ribbon graph from an arrow presentation, we glue discs to each of
the circles and attach edge ribbons to each pair of arrows in such a
way that the arrows form part of a consistent orientation around the
boundary of the edge ribbon.

Here is an example of an arrow presentation.

See more details of the arrow presentation in [Chmutov2009] and [Moffatt2010].

The relation between ribbon graphs and relative
plane graphs is based on the standard notion of a medial graph
(see, for example [Biggs1993]; [Godsil and Royle2001]; [Loebl2010]).

Definition 2.3.

Let $H$ be a planar graph. Its medial graph $M(H)$
is the planar graph whose vertices are the mid-points of the edges
of $H$, and whose edges are given by the following procedure:
whenever two edges are adjacent in some face of $H$, we connect the
corresponding vertices of $M(H)$ by an edge that follows the
boundary of the face. Each vertex of $M(H)$ will be 4-valent. The
medial graph is embedded into the same plane as $H$; each of its
faces corresponds either to a face of $H$ or to a neighborhood of a
vertex of $H$.

The top figure below exemplifies the construction of
the medial graph around an edge of $H$. Here we draw one pair of
opposite edges of $M(H)$ by solid lines and another pair by dotted
lines. The bottom figure shows an example of the entire medial
graph.

Since $M(H)$ is a regular 4-valent planar graph, we may consider it
as an immersion of a number of circles into the plane: if a circle
goes into a vertex of $M(H)$ along some edge of $M(H)$, it continues
to go out of the vertex along the opposite edge of $M(H)$. Then
another pair of edges at this vertex belongs either to the same
circle or to a different one. We draw the edges of one circle by
solid lines and the edges of a different circle by dotted lines. The
number of these circles is denoted by $\delta(H)$. In particular, for
the medial graph of the bottom figure above $\delta(H)=2$. This
immersion of circles has only double points as singularities,
which are points in the plane at which the immersion is two-to-one,
but the tangent directions at this point are distinct.

Construction 2.4.

In the other direction, for a regular 4-valent planar graph $B$ we
can construct a graph $H:=C(B)$ for which the medial graph is equal
to $B$, $M(C(B))=B$. To construct $H$ we consider a black and white
checkerboard coloring of the regions of the complement to $B$ with
the outer region painted white. For any planar 4-valent graph such
coloring does exist. It is given by a parity of the intersection
index of a path connecting a point at infinity with a point in
interior of a region. Then we place a vertex into each black region
and connect two vertices by an edge for each common double point on
their boundaries. This edge is drawn through the corresponding
double point.

Let $R$ be a ribbon
graph and $\widehat{R}$ be its core graph, obtained by forgetting
the ribbon graph structure on the vertices and edges. $\widehat{R}$
embeds naturally into $R$, by placing each vertex of $\widehat{R}$
in the interior of the corresponding vertex disc of $R$, and
connecting these vertices by edges through the corresponding
edge-ribbons of $R$, in such a way that the cyclic order of the
edges around each vertex of $\widehat{R}$ matches the cyclic order
of the edge-ribbons around each vertex disc of $R$. In the same
manner as for planar graphs, we may then construct the medial graph
of $\widehat{R}$ (which we will also denote as $M(R)$ with respect
to this embedding, by placing a vertex at the center of each edge of
$\widehat{R}$ and connecting the vertices of $M(R)$ that belong to
edges which are adjacent in the cyclic ordering around a vertex of
$\widehat{R}$ by an edge that follows the boundary of $R$, and does
not intersect $\widehat{R}$. The construction of $M(R)$ gives an
embedding of this graph into $R$: we will require for convenience
that, in this embedding, the vertex of $M(R)$ corresponding to a
given edge of $\widehat{R}$ lies in the interior of the
corresponding edge-ribbon of $R$. The connected components of $R-M(R)$ are disks and cylinders. The disks correspond to the vertices
of $\widehat{R}$ and the cylinders correspond to the boundary
components of $R.$

The manner in which we draw ribbon graphs suggests to consider a
projection $\pi:R\to{\mathbb{R}}^{2}$ with singularities of two types. The
first occurs when two edge-ribbons cross over each other. The second
occurs when an edge ribbon twists over itself. Away from the
singularities the projection is one-to-one.

The image $B$ of $M(R)$ may then be considered as a regular 4-valent
planar graph whose vertices are divided into two types. The vertices
which are images of vertices of $M(R)$ will be called regular
vertices, and the vertices that arise from the singularities of the
projection will be called 0-vertices. By applying the 2.4, we then obtain a relative planar graph $G:=C(B)$,
whose 0-edges correspond to the 0-vertices of $B$.

Example 2.5.

In this figure the 0-edges of $G$ are drawn as dashed lines.

Of course such a projection always exists for any ribbon graph. In
fact, these projections are easily constructed from an arrow
presentation of a ribbon graph. We consider the circles of the arrow
presentation as disjoint circles in the plane, none of which is
contained in another. The vertex discs are constructed by filling in
these circles. The edge ribbons are constructed in the plane by
first considering arcs connecting the corresponding arrows on each
circle which intersect transversally in the plane, and then taking
sufficiently small neighborhoods of these arcs in the plane. If an
edge ribbon must twist, we incorporate the twist in the ribbon away
from any of the intersections of the arcs.

The constructed relative plane graph $G$ clearly depends on the
projection $\pi$ and on the position of vertices of the medial graph
on the edge-ribbons. However the invariants we will work with will
not be affected by this ambiguity. The figure above shows the
dependence of $G$ on the position of a vertex of the medial graph.

Conversely, from a relative plane graph $G$ we may construct a
ribbon graph $R$. Consider the spanning subgraph $H$ of $G$ whose
edges are the 0-edges of $G$. Construct $M(H)$ as in Sect. 2.2. Consider the medial graph as an immersion of a
collection of $\delta(H)$ circles with clean double points. Each
regular edge of $G$ intersects the planar graph $M(H)$ in two
points. Each of these points has a neighborhood in which the
immersion is an embedding. For each regular edge of $G$, take a
square $I^{2}$ and identify one edge with a neighborhood of an
intersection point in $M(H)$, and identify the opposing edge with a
neighborhood of the second intersection point in $M(H)$, so that the
counterclockwise orientation of the plane and the counterclockwise
orientation of the boundary of $I^{2}$ are compatible. Via the
embedding in a neighborhood of each intersection point, we may pull
these identifications back to the collection of $\delta(H)$ disjoint
circles. The ribbon graph $R$ is then the quotient space obtained by
filling in each of these circles by a disc, and performing the
constructed identifications of these circles with the collection of
squares $I^{2}$ corresponding to the regular edges of $G$.

Example 2.6.

We do not label the pairs of arrows in this example because there is
only one pair.

One can easily see that if $G$ is a relative plane graph constructed
from a ribbon graph $R$ as in previous subsection, then this
construction recovers $R$ from $G$. Also one may notice that there
is a natural one to one correspondence between the edges of $R$ and
the regular edges of $G$.

The Bollobás–Riordan polynomial, originally defined in
[Bollobás and Riordan2002], was generalized to a multivariable polynomial of
weighted ribbon graphs in [Moffatt2008], [Vignes-Tourneret2009]. We will use a sightly more
general doubly weighted Bollobás–Riordan polynomial of a ribbon
graph $R$ with weights $(x_{e},y_{e})$ of an edge $e\in R$.

Definition 2.7.

where the sum runs over all spanning subgraphs $F$, $k(F)$ is the
number of connected components of $F$, $n(F)=|E(F)|-v(F)+k(F)$ is
the nullity of $F$, and $bc(F)$ is the number of boundary
components of $F$.

Definition 2.8.

Let $G$ be a relative plane graph with the distinguished set of
0-edges $H$. We consider spanning subgraphs $F$ of $G$ containing
all 0-edges $H$. Such spanning subgraph can be identified with a
subset of edges of $G\setminus H$. Summing over all such spanning
subgraphs we set

where $\overline{F}=G\setminus(F\cup H)$, $\psi$ is a
block-invariant function on graphs, and $H_{F}$ is the plane graph
obtained from $F\cup H$ by contracting all edges of $F$. Our choice
of $\psi$ is

$\delta(H_{F})$ is the number of circles that immerse to the medial
graph of $H_{F}$.

Remarks.

1.

The relative Tutte polynomial was introduced by Diao and
Hetyei in [Diao and Hetyei2010], who used the notion of activities to
produce the most general form of it. The all subset formula we use
was discovered by a group of undergraduate students (Carnovale,
Dong, Jeffries) at the OSU summer program “Knots and
Graphs” in 2009. However, similar expressions may be traced back to
Traldi ([Traldi2004]) for the non-relative case, and to Chaiken
([Chaiken1989]) for the relative case of matroids.

2.

The function $\psi$ in [Diao and Hetyei2010] can be obtained from ours
by substitution $w=1$.

3.

Another difference with [Diao and Hetyei2010] is that we are using a
doubly weighted version of the relative Tutte polynomial with
weights $(x_{e},y_{e})$ of an edge $e\in G\setminus H$.

4.

In the process of constructing the graph $H_{F}$ by
contracting the edges of $F$ in $F\cup H$, we may come to a
situation when we have to contract a loop. Then the contraction of a
loop actually means its deletion. Since $G$ and $F\cup H$ are plane
graphs, then the graph $H_{F}$ is also embedded into the plane.

5.

While the medial graph of the planar graph $H_{F}$ depends
on the embedding of $H_{F}$ into the plane, the number $\delta(H_{F})$
does not (see [Diao and Hetyei2010]). It depends only on the abstract graph
$H_{F}$.

Theorem 1.

Suppose $R$ is a ribbon graph, and $G$ is a relative plane graph
associated to a projection of $R$. Or, equivalently, assume $G$ is a
relative plane graph and $R$ is the ribbon graph arising from $G$.
Assume that the natural bijection between the edges of $R$ and
regular edges of $G$ preserves the weights.

It is a remarkable consequence of the main theorem that the specialization ($w=\sqrt{\frac{X}{Y}},d=\sqrt{XY}$) of the relative Tutte polynomial does not depend on the various choices made in the construction of the relative plane graph in Sect. 2.3. It is not difficult to describe a sequence of moves on relative plane graphs relating the graphs with different choices of the regular edges. It would be interesting to find such moves for different choices of the projection $\pi$ and, more generally, the moves preserving the relative Tutte polynomial.

2.

The construction of $G$ from $R$ and backward can be generalized to a wider class of projections $\pi$. We can require that only the restriction of $\pi$ to the boundary of $R$ be an immersion with only ordinary double points as singularities. The theorem holds in this topologically more general situation. However, from the point of view of graph theory it is more natural to restrict ourselves to the class of projections which we use.

Our constructions of $G$ from $R$ and $R$ from $G$ in Sects. 2.3 and 2.4 give a bijection between regular
edges of $G$ and the edge-ribbons of $R$. We denote the
corresponding edges by the same letter $e$ for both $e\in G\setminus H$ and for $e\in R$ since this will not lead to confusion. Moreover,
in the theorem we assume that this bijection respects the weights of
the doubly weighted polynomials. The bijection can be naturally
extended to the bijection between spanning subgraphs $F\subseteq G\setminus H$ and $F^{\prime}\subseteq R$ so that the weights of $F$ and
$F^{\prime}$ are equal to each other:

The restriction of the projection $\pi:R\to{\mathbb{R}}^{2}$ from Sect.
2.3 to the spanning ribbon subgraph $F^{\prime}$ is an immersion
of $bc(F^{\prime})$ circles into the plane ${\mathbb{R}}^{2}$. We need to compare this
number of the immersed circles with the number $n(F)+\delta(H_{F})$. To
do this one can check how the number of immersed circles changes
when edges of $F$ are contracted. It is easy to see that the
contraction of a non-loop does not change the number of circles.
But, the contraction of a loop, which is the same as deletion of the
loop, fuses two disjoint circles together, one from the outside of
the loop and one from the inside of the loop. So it reduces the
number of circles by 1. The result of contracting all the edges of
$F$ is the graph $H_{F}$, for which the number of circles will be
$\delta(H_{F})$. Since the number of loops contracted during the process
of contraction is $n(F)$, we have

Consider $F\cup H$ as a spanning subgraph of $G$ and remove the
edges of $H$ from it. Then we get the spanning subgraph $F$. Its
edges are supposed to be contracted, so each connected component of
$F$ gives a vertex of the resulting graph. Now restoring the edges
of $H$ does not change the number of vertices of the graph obtained
by contracting $F$. Thus $v(H_{F})=k(F)$.

Let $G$ be a relative plane graph. The dual of $G$, denoted
$G^{*}$ is formed by taking the dual of $G$ as a plane graph, and
labeling the edges of $G^{*}$ which intersect 0-edges of $G$ as the
0-edges of $G^{*}$. Note that for relative plane graphs $(G^{*})^{*}=G$,
as with usual planar duality.

Theorem 2.

with the correspondence on the edge weights being $x_{e}$=$y_{e^{*}}$,
$y_{e}$=$x_{e^{*}}$, where $e^{*}$ is the edge of $G^{*}$ that intersects
$e$, and $a(G,H)=(|E(G\setminus H)|-v(G))/2+k(G),\,\,b(G)=v(G)/2\ .$

Remarks.

1.

This theorem generalizes the classical relation,
$T_{G}(x,y)=T_{G^{*}}(y,x)$, for the Tutte polynomials of dual plane
graphs to relative plane graphs. The duality theorem for the
Bollobás–Riordan polynomial was found in [Ellis-Monaghan and Sarmiento2011] (see also
[Moffatt2008] and [Chmutov2009]), and for the more general Krushkal’s
polynomial in [Krushkal2011].

2.

The theorem could be proved knowing that the dual of a
relative plane graph corresponds to the dual ribbon graph and using
the Bollobás–Riordan duality result from [Ellis-Monaghan and Sarmiento2011]. However, at
this moment we do not claim this relation and give a direct proof
below. In general, it would be interesting to express the partial
duality of ribbon graphs from [Chmutov2009], [Moffatt2010] in terms of relative
plane graphs.

Proof of the Theorem.

The equality is on monomials of $T_{G,H}$, $T_{G^{*},H^{*}}$ in the edge
weight variables $(x_{e},y_{e})$ which establish the correspondence
between spanning subgraphs $F$ of $G\setminus H$ and $F^{*}$ of
$G^{*}\setminus H^{*}$. Namely, $F^{*}$ consists of those regular edges of
$G^{*}$ which do not intersect the regular edges of $F$.

We prove the equality on monomials for the exponent of $X$. Equality
for $Y$ then follows from duality. The exponent of X on the left is

Now, $|E(F^{*})|=|E(\overline{F})|$ by the subgraph correspondence.
The equality $bc(F_{R})$=$bc{}(F_{R}^{*})$ follows from the fact
that the ribbon graphs $F_{R}$ and $F_{R}^{*}$ have the same boundary. It
can also be seen from the following figures:

In this section we generalize the result of [Diao and Hetyei2010] which extends
the Thistlethwaite theorem to virtual links. Virtual links are
represented by diagrams similar to ordinary knot diagrams, except
some crossings are designated as virtual. Here are some
examples of virtual knots.

Virtual link diagrams are considered up to plane isotopy, the classical Reidemeister moves:

and the virtual Reidemeister moves:

The Kauffman bracket for virtual links is defined in the same way as
for classical links. Let $L$ be a virtual link diagram. Consider two
ways of resolving a classical crossing. The A-splitting,
is obtained by joining the two vertical angles swept out by the
overcrossing arc when it is rotated counterclockwise toward the
undercrossing arc. Similarly, the B-splitting,
is obtained by joining the other two vertical angles. A state
$s$ of a link diagram $L$ is a choice of either an $A$ or
$B$-splitting at each classical crossing. Denote by $\mathcal{S}(L)$ the set
of states of $L$. A diagram $L$ with $n$ crossings has $|\mathcal{S}(L)|=2^{n}$ different states.

Denote by $\alpha(s)$ and $\beta(s)$ the numbers of $A$-splittings and
$B$-splittings in a state $s$, respectively, and by $\delta(s)$ the
number of components of the curve obtained from the link diagram $L$
by splitting according to the state $s\in\mathcal{S}(L)$. Note that
virtual crossings do not connect components.

Definition 6.1.

The Kauffman bracket of a diagram $L$ is a polynomial in
three variables $A$, $B$, $d$ defined by the formula

Note that $[L]$ is not a topological invariant of the
link; it depends on the link diagram and changes with Reidemeister
moves. However, it determines the Jones polynomial $J_{L}(t)$
by a simple substitution:

In 1987 Thistlethwaite ([Thistlethwaite1987]) (see also [Kauffman1988]) proved that
up to a sign and a power of $t$ the Jones polynomial $V_{L}(t)$ of an
alternating link $L$ is equal to the Tutte polynomial
$T_{G_{L}}(-t,-t^{-1})$ of the Tait graph $G_{L}$ obtained from a
checkerboard coloring of the regions of a link diagram.

Kauffman ([Kauffman1989]) generalized the theorem to arbitrary
(classical) links using signed graphs. To virtual links this theorem
was extended in [Chmutov2009], [Chmutov and Pak2007], [Chmutov and Voltz2008] using ribbon graphs. Another
extension, using the relative Tutte polynomial, is due Diao and
Hetyei ([Diao and Hetyei2010]). In their construction the relative plane graph
is the Tait graph of a virtual link diagram whose 0-edges correspond
to virtual crossings. They expressed $[L](A,A^{-1},-A^{2}-A^{-2})$
as a specialization of the relative Tutte polynomial. The whole
Kauffman bracket $[L](A,B,d)$, although not a link invariant, is
of interest as a pure combinatorial invariant of link diagrams. It
turns out that it also can be expressed as a specialization of the
relative Tutte polynomial.

Following [Diao and Hetyei2010], we assign signs to the edges of the Tait graph
$G$ depending on whether the edge connects $A$- or $B$-splitting
regions:

Theorem 3.

Let L be a virtual link diagram, and G the relative plane Tait graph
of Then, under the substitution

There are several other polynomial invariants of graphs on surfaces.
This section is intended to be a guide for the interested reader to
understand how these polynomial invariants are related to each
other, and how our work on the Bollobás–Riordan polynomial and
relative Tutte polynomial fits within this more general context.

One of the most general such polynomials $P_{R}(X,Y,A,B)$ was defined
by Krushkal in [Krushkal2011] in terms of the topology of the
embedding. It generalizes the Bollobás–Riordan polynomial:

$\displaystyle B_{R}(X,Y,Z)=Y^{g}P_{R}(X,Y,YZ^{2},Y^{-1}),$

where $g$ is the genus of the ribbon graph.

A combinatorial polynomial $LV_{R}(x,y,z)$ was defined by Las
Vergnas in [Las1980], [Las Vergnas1999] using matroids of the graph and its dual.
It turns out to be a specialization of the Krushkal polynomial
[Askanazi et al.2013]:

$\displaystyle LV_{R}(x,y,z)=z^{g}P_{R}(x-1,y-1,z^{-1},z).$

The Bollobás–Riordan polynomial was extended to ribbon graphs with
additional structure, arrow structure, in [Bradford et al.2012]. It would be
interesting to define this structure for relative planar graphs and
extend our main theorem to it. Some other polynomial invariants may
be found in [Ellis-Monaghan and Moffatt2015].

The next diagram represents various relations between these
polynomials.

Both the relative Tutte polynomial of [Diao and Hetyei2010] and the Las Vergnas
polynomial of [Las1980], [Las Vergnas1999] may be formulated for matroids. But the
results of [Askanazi et al.2013] (see also the substitutions in the diagram
above) indicate that the Las Vergnas and the Bollobás–Riordan
polynomials are independent. Since the latter polynomial specializes
to the relative Tutte polynomial one should expect that the relative
Tutte and the Las Vergnas polynomials are also independent. This may
signify the existence of a more general matroid polynomial which
would be a matroidal counterpart of the Krushkal polynomial.
Recently this sort of polynomial was found in [Chun et al.2014].